Goal

Theory

What is an Image Histogram?

It is a graphical representation of the intensity distribution of an image.

It quantifies the number of pixels for each intensity value considered.

What is Histogram Equalization?

It is a method that improves the contrast in an image, in order to stretch out the intensity range (see also the corresponding Wikipedia entry).

To make it clearer, from the image above, you can see that the pixels seem clustered around the middle of the available range of intensities. What Histogram Equalization does is to stretch out this range. Take a look at the figure below: The green circles indicate the underpopulated intensities. After applying the equalization, we get an histogram like the figure in the center. The resulting image is shown in the picture at right.

How does it work?

Equalization implies mapping one distribution (the given histogram) to another distribution (a wider and more uniform distribution of intensity values) so the intensity values are spread over the whole range.

To accomplish the equalization effect, the remapping should be the cumulative distribution function (cdf) (more details, refer to Learning OpenCV). For the histogram \(H(i)\), its cumulative distribution \(H^{'}(i)\) is:

\[H^{'}(i) = \sum_{0 \le j < i} H(j)\]

To use this as a remapping function, we have to normalize \(H^{'}(i)\) such that the maximum value is 255 ( or the maximum value for the intensity of the image ). From the example above, the cumulative function is:

Finally, we use a simple remapping procedure to obtain the intensity values of the equalized image: